Theorem madebdayim 33997

Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
madebdayim	⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6788	. . 3 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ dom M )
2		madef 33967	. . . 4 ⊢ M :On⟶𝒫 No
3	2	fdmi 6596	. . 3 ⊢ dom M = On
4	1, 3	eleqtrdi 2849	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ On)
5		fveq2 6756	. . . . . 6 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6		sseq2 3943	. . . . . 6 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
7	5, 6	raleqbidv 3327	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏))
8		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
9	8	sseq1d 3948	. . . . . 6 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
10	9	cbvralvw 3372	. . . . 5 ⊢ (∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏)
11	7, 10	bitrdi 286	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏))
12		fveq2 6756	. . . . 5 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
13		sseq2 3943	. . . . 5 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14	12, 13	raleqbidv 3327	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴))
15		elmade2 33979	. . . . . . . 8 ⊢ (𝑎 ∈ On → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
16	15	adantr 480	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17		elpwi 4539	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝒫 ( O ‘𝑎) → 𝑙 ⊆ ( O ‘𝑎))
18		elpwi 4539	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝒫 ( O ‘𝑎) → 𝑟 ⊆ ( O ‘𝑎))
19	17, 18	anim12i 612	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)))
20		unss 4114	. . . . . . . . . 10 ⊢ ((𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)) ↔ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
21	19, 20	sylib 217	. . . . . . . . 9 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23		simplll 771	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24		dfss3 3905	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎))
25		fveq2 6756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → ( bday ‘𝑦) = ( bday ‘𝑧))
26	25	sseq1d 3948	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑧) ⊆ 𝑏))
27	26	rspccv 3549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
28	27	ralimi 3086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
29		rexim 3168	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
31	30	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
32		elold 33980	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
33	32	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
34		bdayelon 33898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ‘𝑧) ∈ On
35		onelssex 33563	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( bday ‘𝑧) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
36	34, 35	mpan 686	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
37	36	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
38	31, 33, 37	3imtr4d 293	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) → ( bday ‘𝑧) ∈ 𝑎))
39	38	ralimdv 3103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
40	24, 39	syl5bi 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
41	40	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
42	41	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
43		bdayfun 33894	. . . . . . . . . . . . . . . . 17 ⊢ Fun bday
44		oldssno 33972	. . . . . . . . . . . . . . . . . . 19 ⊢ ( O ‘𝑎) ⊆ No
45		sstr 3925	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ∧ ( O ‘𝑎) ⊆ No ) → (𝑙 ∪ 𝑟) ⊆ No )
46	44, 45	mpan2 687	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ No )
47		bdaydm 33896	. . . . . . . . . . . . . . . . . 18 ⊢ dom bday = No
48	46, 47	sseqtrrdi 3968	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ dom bday )
49		funimass4 6816	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun bday ∧ (𝑙 ∪ 𝑟) ⊆ dom bday ) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
50	43, 48, 49	sylancr 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
51	50	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
52	51	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
53	42, 52	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎)
54		scutbdaybnd 33936	. . . . . . . . . . . . 13 ⊢ ((𝑙 <<s 𝑟 ∧ 𝑎 ∈ On ∧ ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
55	22, 23, 53, 54	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56		fveq2 6756	. . . . . . . . . . . . 13 ⊢ ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘(𝑙 |s 𝑟)) = ( bday ‘𝑥))
57	56	sseq1d 3948	. . . . . . . . . . . 12 ⊢ ((𝑙 |s 𝑟) = 𝑥 → (( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑎))
58	55, 57	syl5ibcom 244	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘𝑥) ⊆ 𝑎))
59	58	expimpd 453	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
60	59	ex 412	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
61	21, 60	syl5 34	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
62	61	rexlimdvv 3221	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
63	16, 62	sylbid 239	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) → ( bday ‘𝑥) ⊆ 𝑎))
64	63	ralrimiv 3106	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎)
65	64	ex 412	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎))
66	11, 14, 65	tfis3 7679	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴)
67		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
68	67	sseq1d 3948	. . . 4 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
69	68	rspccv 3549	. . 3 ⊢ (∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴 → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
70	66, 69	syl 17	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
71	4, 70	mpcom 38	1 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  dom cdm 5580   “ cima 5583  Oncon0 6251  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255   No csur 33770   bday cbday 33772   <<s csslt 33902   |s cscut 33904   M cmade 33953   O cold 33954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-bday 33775  df-sslt 33903  df-scut 33905  df-made 33958  df-old 33959

This theorem is referenced by:  oldbdayim  33998  madebday  34007